Question

One month Jose rented 5 movies and 3 video games for a total of $34. The next month he rented 2 movies and 12 video games for a total of $73. Find the rental cost for each movie and each video game.

Answer 1

EXPLANATION

Let's see the facts:

One month:

Rented movies = 5

Rented video games = 3

Total = $34

Next month:

Rented movies = 2

Rented video games = 12

Total = $73

Let's call x to the rental cost for each movie and y to the rental cost for each game, representing the given data on a system of equations give us the following expressions:

(1) 5x + 3y = 34

(2) 2x + 12y = 73

Now, we need to solve this system of equations:

Isolate x for 5x + 3y = 34:

`\mathrm{Subtract\: }3y\mathrm{\: from\: both\: sides}`
`5x+3y-3y=34-3y`

Simplify:

`5x=34-3y`
`\mathrm{Divide\: both\: sides\: by\: }5`
`(5x)/(5)=(34)/(5)-(3y)/(5)`
`Simplify\colon`
`x=(34-3y)/(5)`
`\mathrm{Substitute\: }x=(34-3y)/(5)`
`\begin{bmatrix}2\cdot(34-3y)/(5)+12y=73\end{bmatrix}`
`Simplify\colon`
`\begin{bmatrix}(68+54y)/(5)=73\end{bmatrix}`
`\mathrm{Multiply\: both\: sides\: by\: }5`
`(5\left(68+54y\right))/(5)=73\cdot\: 5`
`\text{Simplify:}`
`68+54y=365`
`\mathrm{Subtract\: }68\mathrm{\: from\: both\: sides}`
`68+54y-68=365-68`
`Simplify\colon`
`54y=297`
`\mathrm{Divide\: both\: sides\: by\: }54`
`(54y)/(54)=(297)/(54)`
`Simplify\colon`
`y=(11)/(2)`
`\mathrm{For\: }x=(34-3y)/(5)`
`\mathrm{Substitute\: }y=(11)/(2)`
`x=(34-3\cdot(11)/(2))/(5)`
`x=(34-(33)/(2))/(5)`
`x=((35)/(2))/(5)`
`x=(35)/(2\cdot\:5)`
`x=(35)/(10)`
`\mathrm{Cancel\: the\: common\: factor\colon}\: 5`
`x=(7)/(2)`
`\mathrm{The\: solutions\: to\: the\: system\: of\: equations\: are\colon}`
`x=(7)/(2),\: y=(11)/(2)`

Now, as x=rental cost for each movie and y= rental cost for each video game, we can conclude:

Rental cost for each movie = $3.5

Rental cost for each video game = $5.5